siddharth said:
I have time, and am working on it. Could you recommend introductory texts in the aforementioned subjects at the advanced undergrad/beginning graduate level?
Manifolds:
William M. Boothby,
An Introduction to Differentiable Manifolds and Riemannian Geometry, Academic Press, 1986. Might look a bit unfriendly but actually very well written.
John M. Lee,
Introduction to Smooth Manifolds, GTM 218, Springer, 2003. See also his other two books, which complement this one. Worthwhile for emphasis on level of structure alone!
Exterior calculus:
Flanders,
Differential Forms with Applications to the Physical Sciences, Dover reprint. The most user friendly introduction, also very nice chapter on matrix Lie groups and the
Maurer-Cartan form.
Spivak,
Calculus on Manifolds is good for some things.
Differential geometry:
Too many books to mention. The five volume book by Spivak has wonderful stuff but I find the notation over-fussy and off-putting.
Chris J. Isham,
Modern Differential Geometry for Physicists. A nice short book, not unsuitable as a supplementary text for math students.
Lie theory:
Bump, op cit.
The little book by Segal, Carter, and McDonald, LMS student text Vol. 32, is too sketchy to use as a textbook but is highly readable and very inspiring. (There is a mistake in the discussion of Penrose's description of the night sky via the Lorentz group.)
The books by Vadarajan, Humphreys, and Knapp are all excellent but maybe not the best first text. The book by Gilmore,
Lie Groups, Lie Algebras, and Some of Their Applications is notable for an unique mathematical symbol (a picture of banannas which appears as an argument), and has many convenient tables and is a bit more historical than some. The books by Hermann are idiosyncratic but intriguing, time permitting.
If you have any more questions, don't take it personally if I don't reply (lack of time, etc). Good luck!
